Re: scalling of poisson distribution


"Nick" <tulse04-news1@xxxxxxxxxxx> wrote in message
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"Daniel Oberhoff" <daniel.oberhoff@xxxxxxxxxxxxxxxxx> wrote in message
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Hi all,

I am having some trouble wrapping my head around the poisson
distribution. It is supposed to be a good model for firing rate
distributions of neurons which makes sense to me. But rates are always
expressed in some unit (per second, minute, whatever), but the choice of
unit should not alter the distribution. In the case of the poisson
distribution however it seems to. Or in other words:

p = poiss(n,l) (where l = <x>)

does not scale according to:

p -> c * l, n -> c * n, l -> c * l

which means the distribution is different depending on how long I
count??? Even more confusng since I read on Wikipedia that the poisson
process is memoryless, so doubling the time interval I should just
expect double the occurences???

can anyone explain this to me?

It is a long time since I did this, but if the mean is lambda for period
t, then the mean is lambda for period 2t

The probability (x,2*lambda) = (exp(-lambda)*lambda^x)/x factorial

and Sum of the probability (x,2*lambda) (x=0,1,...) is 1

I think that the fact that the distribution is discrete rather than
continuous comes into it. After all, if we take it the other way round, if
we have x=0,1,2,... in unit time t and we then halve the unit to t/2, we
can't divide the x values by 2. We are still looking at discrete values of
x. If it was a continuous variable then we could truly scale the
distribution.

In fact, see the contribution in the other thread where it is stated that
doctor's visits per day are Poisson, whereas doctor's visits per year are
binomial. You could say that that is just a matter of scale.

Nick


.

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